ITÔ-SKOROHOD STOCHASTIC EQUATIONS
AND APPLICATIONS TO FINANCE
CIPRIAN A. TUDOR
Received 6 November 2003 and in revised form 8 June 2004
We prove an existence and uniqueness theorem for a class of Itô-Skorohod stochastic
equations. As an application, we introduce a Black-Scholes market model where the price
of the risky asset follows a nonadapted equation.
1. Introduction
The introduction of the anticipating (or Skorohod) integral in [8] and of the anticipating stochastic calculus in [7] has opened the question of solving anticipating stochastic
differential equations. In general, the existence and uniqueness of the solution for these
equations is not known. The difficulty of solving such equations is due to the fact that the
classical method of Picard iterations cannot be applied because the mean square formula
for the Skorohod integral involves the Malliavin derivation in a such way that we cannot
find “closed” formulas. Only in few particular cases do some results exist; see, for example, [1, 2, 3]. We have recently proved in [9] that the set of Skorohod integrals coincides
with a set of integrals of Itô type. In the present work, using this correspondence between
Skorohod integrals and Itô-Skorohod integrals, we introduce a class of anticipating equations (called Itô-Skorohod equations) that can be solved using standard techniques. As an
application, we introduce a market model where the price of the risky asset follows such
an equation with a random initial condition (the price at the transaction time). We prove
that our model is complete and has no arbitrage opportunities and we derive a BlackScholes formula when the initial price of the risky asset is given by a standard normal
random variable.
We organized the paper as follows. Section 2 contains some preliminaries on the anticipating stochastic calculus. In Section 3, we define the class of Itô-Skorohod equations
and we prove the existence and uniqueness of the solution. In Section 4, we introduce a
market model with price dynamics following an Itô-Skorohod equation and we obtain a
Black-Scholes option valuation formula and the expression of the replicant portfolio.
Copyright © 2004 Hindawi Publishing Corporation
Journal of Applied Mathematics and Stochastic Analysis 2004:4 (2004) 359–369
2000 Mathematics Subject Classification: 60H05, 60H07
URL: http://dx.doi.org/10.1155/S1048953304311044
360
Itô-Skorohod stochastic equations
2. Preliminaries
We start with some elements of the Malliavin calculus. We refer to [6] for a complete
presentation of this topic. Let (W(t))t∈[0,1] be a standard Wiener process on the canonical
Wiener space (Ω, F,P) and let (Ft )t∈[0,1] be the filtration generated by W. A functional of
the Brownian motion of the form
F = f W t1 ,...,W tn ,
(2.1)
with t1 ,...,tn ∈ [0,1] and f ∈ Cb∞ (Rn ), is called a smooth random variable and this class
is denoted by ᏿. The Malliavin derivative is defined on ᏿ as
Dt F =
n
∂f ∂xi
i =1
t ∈ [0,1],
W t1 ,...,W tn 1[0,ti ] (t),
(2.2)
if F has the form (2.1). The operator D is closable and can be extended to the closure of
᏿ with respect to the seminorm
p
F k,p = E|F | p +
k
p
ED( j) F 2
L ([0,1]) ,
j =1
(2.3)
where D(i) denotes the ith iterated derivative. The adjoint of D is denoted by δ and is
called the Skorohod integral. That is, δ is defined on its domain
1
Dom(δ) = u ∈ L2 [0,1] × Ω / E
0
us Ds F ds
≤ C F L2 (Ω)
(2.4)
and is given by the duality relationship
E Fδ(u) = E
1
0
u ∈ Dom(δ), F ∈ ᏿.
us Ds F ds,
(2.5)
Recall that the variance of the Skorohod integral is
E δ 2 (u) = E
1
0
u2α dα + E
11
0
0
Dβ uα Dα uβ dαdβ.
(2.6)
By Lk,p , we denote the set L2 ([0,1]; Dk,p ), for k ≥ 1 and p ≥ 2, and we note that Lk,p is a
subset of the domain of δ. The following version of the Ocone-Clark formula was given
in [7]:
F = E F/ F[s,t]c +
t
s
E Dα F/ F[α,t]c dW(α),
for F ∈ D1,2 .
(2.7)
Ciprian A. Tudor 361
We will need the integration-by-parts formula
Fδ(u) = δ(Fu) +
[0,1]
Ds Fus ds
(2.8)
if all above terms are defined. Recall also that if F is a random variable, Malliavin differentiable, and measurable with respect to a σ-algebra FA , A ∈ Ꮾ(R), then
on Ac × Ω.
DF = 0,
(2.9)
We define, for k ≥ 1 and p ≥ 2, the sets of processes
t
t
ᏹk,p = X = Xt
ᏺ
k,p
t ∈[0,1] , Xt =
= Y = Yt
t ∈[0,1] ,
Yt =
0
0
us dWs , u ∈ Lk,p ,
E vs / F[s,t]c dWs , v ∈ L
k,p
(2.10)
.
We will refer to the elements of ᏺk,p as Itô-Skorohod integral processes and to the elements of ᏹk,p as Skorohod integral processes. It has been proved in [9] that for sufficiently
regular integrands, the two classes coincide. As a consequence, to study Skorohod integral processes, it suffices to study Itô-Skorohod integral processes, which have two intert
esting properties. Firstly, note that the integral Yt = 0 E[uα / F[α,t]c ]dWα exists even for
u ∈ L2 ([0,1] × Ω) and has similarities to a classical Itô integral. Observe, by (2.6), that
this integral is an “isometry”:
t
E
0
E uα / F
[α,t]c
2
dWα
Secondly, if we define, for every λ ≤ t, Ytλ =
is an F(λ,t]c -martingale and we have
=E
λ
0
lim Ytλ = Yt
t
0
E uα / F[α,t]c
2
dα.
E[uα / F[α,t]c ]dWα , then the process (Ytλ )λ≤t
a.s. and in L2 .
λ→t, λ≤t
(2.11)
(2.12)
We will now define the stochastic integral with respect to Itô-Skorohod integral processes.
Definition 2.1. Let u,v ∈ L2 ([0,1] × Ω) be adapted processes and consider more Yt = Y0 +
t
t
0 E[uα / F[α,t]c ]dWα + 0 E[vα / F[α,t]c ]dα. By definition, for any adapted square integrable
process X,
t
0
Xs dYs :=
t
0
Xs ds Yts ,
(2.13)
where
Ytλ
= Y0 +
λ
0
E uα / F[α,t]c dWα +
λ
0
E vα / F[α,t]c dα
(2.14)
and the integral on the right-hand side of (2.13) is understood in the semimartingale
sense.
362
Itô-Skorohod stochastic equations
3. Itô-Skorohod stochastic equations
In this section, we state and prove an existence and uniqueness theorem for a class of
anticipating stochastic differential equations using the method of Picard iterations. It is
known that in the anticipating stochastic calculus, this method cannot be applied because
the formula of the mean square of the Skorohod integral involves the Malliavin derivative and one cannot find “closed” formulas. We define here a new class of anticipating
equations, located “between” Itô and Skorohod equations, that can be solved by classical
techniques. Consider the following stochastic differential equation:
Xt = Z +
t
0
σ s, E Xs / F[s,t]c dWs +
t
0
b s,Xs ds.
(3.1)
Note that the stochastic integral above is a Skorohod integral since the integrand is not
adapted and the initial condition is anticipating. The solution will also be anticipating.
In what follows, the coefficients σ(t,x),b(t,x) : [0,1] × R → R are given and satisfy the
following standard conditions.
(H1) (Measurability): σ and b are jointly measurable in (t,x) ∈ [0,1] × R.
(H2) (Lipschitz condition): there exists a D > 0 such that for all t ∈ [0,1] and x ∈ R,
σ(t,x) − σ(t, y) + b(t,x) − b(t, y) ≤ D|x − y |.
(3.2)
(H3) (Linear growth condition): there exists a C > 0 such that for all t ∈ [0,1] and
x ∈ R,
σ(t,x)2 + b(t,x)2 ≤ C 2 1 + |x|2 .
(3.3)
We also make a hypothesis concerning the initial value Z.
(H4) Z is a random variable with E|Z |2 < ∞.
A square integrable process that satisfies a.s. (3.1) is called a solution of (3.1). For given
coefficients σ and b, any solution X will depend on the initial value Z. We will say that the
solution is unique if, for every t ∈ [0,1], P(Xt1 = Xt2 ) = 1 for any two solutions X 1 and X 2
with the same initial condition.
We start by proving the existence and the uniqueness of the solution of (3.1).
Theorem 3.1. Under assumptions (H1), (H2), (H3), and (H4), stochastic equation (3.1)
has a unique solution Xt on [0,1] with
2
sup EXt < ∞.
(3.4)
0≤t ≤1
Proof. Throughout this proof, K will denote a generic constant depending only on D and
(0)
E|Z |2 . We consider the usual Picard iterations Xt = Z and
Xt(n+1) = Z +
t
0
σ s, E Xs(n) / F[s,t]c dWs +
t
0
b s,Xs(n) ds.
(3.5)
Ciprian A. Tudor 363
We first prove the existence of the solution. We have, from (2.6), (H3), and Hölder’s
inequalities, that
t
2
t
2
(1)
(0) 2
c
EXt − Xt ≤ 2E σ s, E Z/ F[s,t] dWs + 2E b(s,Z)ds
≤ 2E
t
0
≤ 2C 2 E
0
σ s, E Z/ F[s,t]c 2 ds + 2t E
t
t
0
0
b(s,Z)2 ds
2 1 + E Z/ F[s,t]c ds + 2tC 2 E
0
t
0
(3.6)
1 + |Z |2 ds
≤ Kt.
Using the same arguments and condition (H4), we obtain
(n+1)
EXt
t
2
(n)
(n−1)
(n) 2
c
c
− Xt ≤ 2E
σ
s,
E
X
/
F
σ
s,
E
X
/
F
dW
−
[s,t]
[s,t]
s
s
s
0
t
2
(n)
(n−1)
+ 2E
b s,Xs − b s,Xs
ds
0
t
(n)
2
≤ 2E
σ s, E Xs / F[s,t]c − σ s, E Xs(n−1) / F[s,t]c
ds
0
+ 2t E
t
0
b s,Xs(n) − b s,Xs(n−1)
≤ 2D2 (1 + t)
t
0
2
(3.7)
ds
2
EXs(n) − Xs(n−1) ds.
By induction, one can show that there exists K > 0 such that for all t ∈ [0,1] and n ≥ 1,
(n+1)
EXt
(Kt)n+1
(n) 2
− Xt ≤
.
(3.8)
(n + 1)!
Relation (3.8) and standard arguments imply the convergence, in L2 (Ω), of the successive
(n+1)
(n)
approximations Xt(n) to a limit Xt defined by Xt = Z + ∞
− Xt ).
n=0 (Xt
2
To prove that X is a solution, we take the L (Ω)-limit in (3.5) as n → ∞. Obviously,
t
2E
0
≤K
t
E
0
σ s, E Xs(n) / F[s,t]c
t
0
2
dWs − σ s, E Xs / F[s,t]c
2
EXs(n) − Xs ds −→n→∞ 0,
2
b s,Xs(n) − b s,Xs ds
≤K
t
0
(3.9)
2
EXs(n) − Xs ds −→n→∞ 0.
364
Itô-Skorohod stochastic equations
The uniqueness of the solution is given by Gronwall’s lemma since for any two solutions
X, Y with the same initial condition and for every t ∈ [0,1], we have
t
2
EXt − Yt ≤ K
0
2
EXs − Ys ds.
(3.10)
Concerning bound (3.4), we will only note that standard techniques apply (see, e.g., [4]).
Remark 3.2. We define the following stochastic differential equation:
Xt = E Z/ Ftc +
t
0
σ s, E Xs / F[s,t]c dWs +
t
0
b s, E Xs / F[s,t]c ds.
(3.11)
Following the lines of the proof of Theorem 3.1, one can show that (3.11) admits a unique
solution X with sup0≤t≤1 E|Xt |2 < ∞.
In the particular case of linear coefficients, one can explicitly obtain the solution of
(3.11).
Corollary 3.3. Let σ,b ∈ R and X0 ∈ L2 (Ω). Consider the equation
Xt = E X0 / Ftc +
t
0
σ E Xs / F[s,t]c dWs +
t
0
bE Xs / F[s,t]c ds.
(3.12)
.
(3.13)
Then the unique solution of (3.12) is given by
Xt = E X0 / Ftc eσWt +(b−σ
Proof. Denote Mt = eσWt +(b−σ
2 /2)t
2 /2)t
. Then Mt satisfies the equation
Mt = 1 +
t
0
t
σMs dWs +
0
bMs ds,
(3.14)
and using (2.8) and (2.9), we obtain
Xt = E X0 / Ftc Mt
= E X0 / Ftc +
= E X0 / F
t
0
[s,t]c
= E X0 / Ftc +
t
+
t
0
σ E X0 / Ftc Ms dWs +
0
t
0
σ E E X0 / F Ms / F
sc
σ E Xs / F[s,t]c dWs +
[s,t]c
t
0
bE X0 / Ftc Ms ds
dWs +
t
0
bE E X0 / Fsc Ms / F[s,t]c ds
bE Xs / F[s,t]c ds.
(3.15)
4. Black-Scholes model driven by Itô-Skorohod stochastic differential equations
We introduce, in this section, a market model with price dynamics following an ItôSkorohod stochastic equation. As usual, we will consider two assets on the probability
Ciprian A. Tudor 365
t
space (Ω, F,P,(Ft )t∈[0,1] ): the safe investment A = (At )t∈[0,1] satisfying At = 1 + r 0 As ds
and the risky asset S = (St )t∈[0,1] with price dynamics following the stochastic differential
equation
t
St = E S0 / Ftc +
0
σ E Ss / F[s,t]c dWs +
t
0
bE Ss / F[s,t]c ds.
(4.1)
.
(4.2)
Clearly, At = ert and Corollary 3.3 implies that
St = E S0 / Ftc eσWt +(b−σ
2 /2)t
The value of the portfolio at the instant t is defined by
Vt = ht At + Ht St ,
(4.3)
where the components h,H ∈ L2 ([0,1] × Ω) are adapted to the Brownian filtration and
represent the quantities of the safe asset and the risky asset at the instant t.
We say that the portfolio (ht ,Ht )t∈[0,1] is self-financing if
Vt = E V0 / Ftc +
t
0
t
hs dAs +
0
Hs dSs ,
(4.4)
where the differential dS is understood in the sense of Definition 2.1.
Remark 4.1. Note that Definition 2.1 can be used although the initial value depends on t
because, by the Ocone-Clark formula (2.7), we can write
E S0 / F
tc
= S0 −
t
0
E Ds S0 / F[s,t]c dWs .
(4.5)
In other words, the self-financing condition (4.4) can be written as
Vt = E V0 / Ftc +
t
+
0
t
0
hs rers ds −
t
0
Hs σ E Ss / F[s,t]c dWs +
Hs E Ds S0 / F[s,t]c dWs
t
0
(4.6)
bHs E Ss / F[s,t]c ds.
In the following, we will denote by S̃t = e−rt St the discounted risky asset price. A necessary and sufficient condition for the portfolio, to be self-financing, is given in the next
result.
Proposition 4.2. Assume that h,H ∈ L2 ([0,1] × Ω) and let the process V be given by (4.3).
Denote Ṽt = e−rt Vt .Then the portfolio is self-financing if and only if
Ṽt = E V0 / Ftc +
t
0
Hs dS̃s
for every t ∈ [0,1].
(4.7)
366
Itô-Skorohod stochastic equations
Proof. Suppose that V satisfies (4.4). Define, for every λ ∈ [0,t],
Vλ,t = E V0 / Ftc +
λ
+
0
λ
0
hs rers ds −
λ
0
Hs σ E Ss / F[s,t]c dWs +
Hs E Ds S0 / F[s,t]c dWs
λ
0
(4.8)
bHs E Ss / F[s,t]c ds.
It is not difficult to check that Vλ,t = E(Vλ / F[λ,t]c ).
We can write Itô’s formula for e−rλ Vλ,t since, for fixed t, the process (Vλ,t )λ∈[0,t] is a
F[λ,t]c -semimartingale. It holds, taking the limit (a.s. or in L2 ) as λ → t, that
t
Ṽt = E V0 / Ftc +
t
+
0
0
= E V0 / Ftc −
t
0
t
t
0
e−rs σHs E Ss / F[s,t]c dWs +
+
e−rs hs rers ds −
e−rs Hs E Ds S0 / F[s,t]c dWs
t
0
e−rs bHs E Ss / F[s,t]c ds +
t
0
Vs,t − re−rs ds
(4.9)
e−rs Hs E Ds S0 / F[s,t]c dWs
0
e−rs σHs E Ss / F[s,t]c dWs +
t
0
e−rs (b − r)Hs E Ss / F[s,t]c ds.
On the other hand, writing Itô’s formula for e−rλ Sλ,t with
Sλ,t = E S0 / Ftc +
λ
0
σ E Ss / F[s,t]c dWs +
λ
0
bE Ss / F[s,t]c ds = E Sλ / F[λ,t]c ,
(4.10)
we get
S̃t = E S0 / Ftc +
t
0
e−rs σ E Ss / F[s,t]c dWs +
t
0
e−rs (b − r)E Ss / F[s,t]c ds.
(4.11)
Identity (4.7) follows from (4.9) and the above equation using Definition 2.1. The proof
of the necessary part is not more difficult.
Let T be the exercise time. In the classical Black-Scholes settings, to prove the nonexistence of arbitrage, it suffices to exhibit a probability measure equivalent to P under which
the discounted price S̃ is a martingale. In our case, we have the following.
Proposition 4.3. The unique probability measure P̃ equivalent to P under which the process S̃t /E(S0 / Ftc ) is a martingale is given by the Radon-Nikodym derivative
r −µ
1 (r − µ)2
dP̃
= exp
T
WT −
dP
σ
2 σ2
P-a.s.
(4.12)
Under probability P̃, the process W̃t = Wt + ((b − r)/σ)t is a standard Brownian motion and
the discounted price S̃ satisfies the equation
S̃t = E S̃0 / Ftc +
t
0
σ E S̃s / F[s,t]c dW̃s .
(4.13)
Ciprian A. Tudor 367
Proof. Denote Zt = S̃t /E(S0 / Ftc ) = e−rt eσWt +(b−σ /2)t . It is well known that there exist the
unique probability P̃ and the Brownian motion W̃ as above and it holds that Zt = 1 +
t
0 σZs d W̃s . Taking into account that the natural filtrations of W and W̃ coincide, we get
2
S̃t = E S̃0 / Ftc Zt = E S̃0 / Ftc +
t
0
σ E S̃0 / Ftc Zs dW̃s = E S̃0 / Ftc +
t
0
σ E S̃s / F[s,t]c dW̃s .
(4.14)
Remark 4.4. Note that, by Corollary 3.3, we have S̃t = E(S̃0 / Ftc )eσ W̃t −(σ /2)t . Also, an immediate consequence of Propositions 4.2 and 4.3 is the fact that the market is complete
and has no arbitrage opportunities.
2
Consider VT = (ST − K)+ the payoff function of the European call option with exercise
time T and strike price K. Denote by Ẽ the expectation with respect to P̃ and by D̃ the
Malliavin derivative with respect to W̃. By formulas (4.7) and (4.13), we have
Ṽt = Ẽ V0 / Ftc −
t
0
Hs Ẽ D̃s S0 / F[s,t]c dW̃s +
t
0
σHs Ẽ S̃s / F[s,t]c dW̃s .
(4.15)
Taking the conditional expectation with respect to the σ-algebra Ft , we obtain
Ẽ Ṽt / Ft = Ẽ V0 −
t
0
Hs Ẽ D̃s S0 / Fs dW̃s +
t
0
σHs Ẽ S̃s / Fs dW̃s .
(4.16)
Therefore, the process (Ẽ(Ṽt / Ft ))t∈[0,1] is a martingale and, for every t ≤ T, it holds that
Ẽ(Ṽt / Ft ) = Ẽ(ṼT / Ft ) or
Ẽ Vt / Ft = Ẽ e−r(T −t) VT / Ft .
(4.17)
We have the following option valuation Black-Scholes formula.
Proposition 4.5. Assume that the terminal value is given by VT = f (ST ) with f (x) =
(x − K)+ and the initial price of the risky asset is S0 = W1 + c, where c is a positive constant.
Then
Ẽ Vt / Ft = G t,
St
,
W1 − Wt + c
(4.18)
where
1
G(t,x) = 2π(1 − T)
(4.19)
2
2
× x
e−u /(1−T) (u + c)N d1 du − Ke−r(T −t) e−u /(1−T) N d2 du
R
R
368
Itô-Skorohod stochastic equations
with
ln K/x(u + c) + r + σ 2 /2 (T − t)
√
,
σ T −t
d2 = d2 (x,u) = d1 − σ T − t,
d1 = d1 (x,u) =
√
and N(d) = (1/ 2π)
d
−∞ e
−x2 /2
(4.20)
dx.
Proof. Using the fact that the increments of the Wiener process are independent on disjoint intervals, the Markov property and (4.17) imply that
Ẽ Vt / Ft = Ẽ e−r(T −t) f ST / Ft
2
2
= Ẽ e−r(T −t) f eσWt +(r −σ /2)t Ẽ S0 / FT c eσ(WT −Wt )+(r −σ /2)(T −t)
(4.21)
St
,
= G t, Ẽ S0 / Ftc
where
G(t,x) = e−r(T −t) Ẽ f xE S0 / FT c eσ(WT −Wt )+(r −σ
2 /2)(T −t)
.
(4.22)
Since E(S0 / FT c ) = W1 − WT + c, by the joint normal distribution of (W1 − WT ,WT − Wt ),
G(t,x) = 1
2π(1 − T)
×
2
R
e−u /2(1−T) e−r(T −t)
2π(T − t)
R
f x(u + c)eσv+(r −σ
2 /2)(T −t)
e −v
2 /2(T −t)
dv du.
(4.23)
We refer to classical arguments (see [5]) to get
e−r(T −t)
2π(T − t)
R
f x(u + c)eσv+(r −σ
2 /2)(T −t)
e −v
2 /2(T −t)
dv
= x(u + c)N d1 (x,u) − Ke−r(T −t) N d2 (x,u) ,
(4.24)
where d1 , d2 are given by (4.20) and the conclusion follows.
Since the market is complete, every bounded contingent claim is attainable. Therefore, it is of importance to find the expression of the replicant portfolio. This is given in
Proposition 4.6.
Proposition 4.6. Under the hypothesis of Proposition 4.5 and denoting g(t,x) =
e−rt G(t,ert x), the replicant portfolio is given by
S̃
Ht = cσ t − 1
Ẽ S0 / Ftc
−1
σ
∂g
S̃
t, t ,
∂x
Ẽ S0 / Ftc
St
S̃t
.
− cHt ht = G t,
W1 − Wt + c
Ẽ S0 / Ftc
(4.25)
Ciprian A. Tudor 369
Proof. Denote Mt = S̃t / Ẽ(S0 / Ftc ). We utilize the classical procedure to determine the unknown quantities h and H. We have that
Ẽ Ṽt / Ft = e
−rt
=e
−rt
Ẽ Vt / Ft = e
−rt
St
Ẽ S0 / Ftc
G t, S̃
G t,e t = e−rt G t,ert Mt
Ẽ S0 / Ftc
(4.26)
rt
with G a C ∞ function on [0,T) × R. Writing Itô’s formula for g(t,Mt ), we obtain
t
∂g u,Mu Mu dW̃u
∂x
0
t
∂g 1 t ∂2 g +
u,Mu du +
u,Mu σ 2 Mu2 du.
2
2 0 ∂x
0 ∂t
g t,Mt = g 0,M0 +
σ
(4.27)
Note first that the bounded variation part is zero. On the other hand, by (4.7),
Ẽ Ṽt / Ft = Ẽ V0 −
t
0
Hs E D̃s S0 / Fs +
t
0
σHs Ẽ S̃s / Fs dW̃s .
(4.28)
By (4.27) and (4.28), the natural candidate for H satisfies σ(∂g/∂x)(s,Ms )Ms =
σHs Ẽ(S̃s / Fs ) and since Ẽ(S̃s / Fs ) = Ẽ(S0 )Ms , we obtain relation (4.25).
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Ciprian A. Tudor: Laboratoire de Probabilités et Modèles Aléatoires, Université de Paris 6, 4 Place
Jussieu, 75252 Paris Cedex 5, France
E-mail address: tudor@ccr.jussieu.fr